Question

find the lengths of w, x, y, and z shown in the figure below if xy = 63. round your answers to the nearest tenth. note that the figure is not drawn to scale. 22 y w z x 12 44° w = 8.3 x = 8.6 y = 7.3 z = 23.2

Explanation:

Step1: Analyze the smaller right triangle (with hypotenuse 12 and angle 44°)

For the smaller right triangle, we can use trigonometric ratios. Let's find \( w \) and \( x \) first.
- To find \( w \) (adjacent to 44°), use \( \cos(44^\circ)=\frac{w}{12} \). So \( w = 12\cos(44^\circ) \approx 12\times0.7193 \approx 8.63 \approx 8.6 \) (wait, but the given \( w \) is 8.3? Wait, maybe I mixed up. Wait, no, maybe the angle is at the bottom. Wait, the smaller triangle has hypotenuse 12, angle 44° at the bottom. So:
- \( \sin(44^\circ)=\frac{w}{12} \)? No, wait, the right angle is between \( x \) (vertical) and \( w \) (horizontal). So the angle at the bottom is 44°, so:
- Opposite to 44°: \( w \), adjacent: \( x \), hypotenuse: 12.
So \( \sin(44^\circ)=\frac{w}{12} \Rightarrow w = 12\sin(44^\circ) \approx 12\times0.6947 \approx 8.336 \approx 8.3 \) (matches the given \( w = 8.3 \)).
\( \cos(44^\circ)=\frac{x}{12} \Rightarrow x = 12\cos(44^\circ) \approx 12\times0.7193 \approx 8.63 \approx 8.6 \) (matches \( x = 8.6 \)).

Step2: Find \( y \) (since \( xy = 63 \))

We know \( x \approx 8.6 \), so \( y = \frac{63}{x} \approx \frac{63}{8.6} \approx 7.325 \approx 7.3 \) (matches \( y = 7.3 \)).

Step3: Analyze the larger right triangle (with horizontal side 22, vertical side \( x + y = 8.6 + 7.3 = 15.9 \)?) Wait, no, the larger triangle has horizontal side 22, vertical side \( y \)? Wait, no, the larger triangle: vertical side is \( y \), horizontal side 22, hypotenuse \( z \), and the angle at the bottom is the same 44° (since the two triangles are similar? Wait, yes, they are similar because both are right triangles with the same acute angle. So the larger triangle: horizontal side 22, vertical side \( y \)? No, wait, the larger triangle's vertical side is \( y \)? Wait, no, the total vertical length is \( x + y \)? Wait, no, the problem says \( xy = 63 \), and we found \( x \approx 8.6 \), \( y \approx 7.3 \), so \( x + y \approx 15.9 \)? Wait, no, maybe the larger triangle has horizontal side 22, vertical side \( y \)? No, let's check similarity. Since both triangles are right-angled and share the acute angle at the bottom, they are similar. So the ratio of sides should be equal.

- Smaller triangle: horizontal \( w = 8.3 \), vertical \( x = 8.6 \), hypotenuse 12.
- Larger triangle: horizontal 22, vertical \( y \)? No, wait, larger triangle: horizontal 22, vertical \( (x + y) \)? No, the larger triangle's vertical side is \( y \)? Wait, no, the figure shows the larger triangle has horizontal side 22, vertical side \( y \), and hypotenuse \( z \), with the same angle 44°. So:
- \( \tan(44^\circ)=\frac{22}{(x + y)} \)? No, wait, \( \tan(44^\circ)=\frac{22}{z_{\text{vertical}}} \)? Wait, no, the larger triangle: horizontal side 22, vertical side \( (x + y) \)? Wait, no, the vertical segment is split into \( x \) (smaller triangle) and \( y \) (larger triangle's vertical part? No, the figure shows the larger triangle is above the smaller one? Wait, the figure: top triangle (larger) has horizontal side 22, vertical side \( y \), right angle. Bottom triangle (smaller) has horizontal side \( w \), vertical side \( x \), right angle. The angle at the bottom is 44° for both. So the two triangles are similar (AA similarity: right angle and 44° angle). So the ratio of corresponding sides:
- Smaller triangle: horizontal \( w \), vertical \( x \), hypotenuse 12.
- Larger triangle: horizontal 22, vertical \( y \), hypotenuse \( z \).
So \( \frac{w}{22}=\frac{x}{y}=\frac{12}{z} \). Wait, but we know \( xy = 63 \), \( w \approx 8.3 \), \( x \approx 8.6 \), so \( y = 63 / 8.6 \approx 7.3 \). Then check similarity: \( \frac{w}{22}=\frac{8.3}{22}\approx0.377 \), \( \frac{x}{y}=\frac{8.6}{7.3}\approx1.178 \) – that's not equal. So maybe the larger triangle's vertical side is \( x + y \)? Wait, no, the right angle in the larger triangle is at the top, so vertical side is \( y \), horizontal side 22, angle at the bottom is 44°, so \( \tan(44^\circ)=\frac{22}{y} \)? No, \( \tan(44^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{22}{y} \Rightarrow y = \frac{22}{\tan(44^\circ)} \approx \frac{22}{0.9657} \approx 22.78 \), which doesn't match. Wait, maybe I got the triangles reversed. Let's re-express:

Wait, the smaller triangle: right angle between \( x \) (vertical) and \( w \) (horizontal), angle 44° at the bottom, hypotenuse 12. So:
- \( \sin(44^\circ) = \frac{w}{12} \Rightarrow w = 12\sin(44^\circ) \approx 8.3 \) (correct).
- \( \cos(44^\circ) = \frac{x}{12} \Rightarrow x = 12\cos(44^\circ) \approx 8.6 \) (correct).

Larger triangle: right angle at the top, horizontal side 22, vertical side \( y \), angle at the bottom is 44°, so:
- \( \tan(44^\circ) = \frac{22}{(x + y)} \)? No, the vertical side of the larger triangle is \( y \), and the total vertical length from bottom to top is \( x + y \)? Wait, no, the problem says \( xy = 63 \), so \( x \) and \( y \) are two segments of the vertical line. So total vertical length is \( x + y \), but the larger triangle has horizontal side 22, vertical side \( y \)? No, the larger triangle's vertical side is \( y \), horizontal side 22, so \( \tan(44^\circ) = \frac{22}{y} \Rightarrow y = 22 / \tan(44^\circ) \approx 22 / 0.9657 \approx 22.8 \), which is not 7.3. So maybe the larger triangle's vertical side is \( x \), and horizontal side 22? No, that doesn't fit. Wait, maybe the two triangles are similar, so \( \frac{w}{22} = \frac{12}{z} = \frac{x}{(x + y)} \)? No, the problem states \( xy = 63 \), so \( x \) and \( y \) are the two parts of the vertical segment. So \( x + y \) is the total vertical length. Let's use the smaller triangle to find \( x \) and \( w \), then find \( y \) from \( xy = 63 \), then find \( z \) using the larger triangle.

We have:
- From smaller triangle: \( x = 12\cos(44^\circ) \approx 8.6 \), \( w = 12\sin(44^\circ) \approx 8.3 \).
- Then \( y = 63 / x \approx 63 / 8.6 \approx 7.3 \).
- Now, the larger triangle has horizontal side 22, vertical side \( y \)? No, the larger triangle has horizontal side 22, vertical side \( (x + y) \)? Wait, no, the larger triangle's vertical side is \( y \), and the angle at the bottom is 44°, so \( \tan(44^\circ) = \frac{22}{y} \)? No, \( \tan(44^\circ) = \frac{\text{horizontal}}{\text{vertical}} = \frac{22}{y} \Rightarrow y = 22 / \tan(44^\circ) \approx 22 / 0.9657 \approx 22.8 \), which is not 7.3. So maybe the larger triangle's vertical side is \( x \), and horizontal side 22? No, \( x \approx 8.6 \), \( 22 / 8.6 \approx 2.55 \), not \( \tan(44^\circ) \approx 0.9657 \). Wait, maybe the larger triangle has hypotenuse \( z \), horizontal side 22, vertical side \( x + y \)? Wait, \( x + y = 8.6 + 7.3 = 15.9 \). Then \( z = \sqrt{22^2 + 15.9^2} \approx \sqrt{484 + 252.81} \approx \sqrt{736.81} \approx 27.1 \), which doesn't match the given \( z = 23.2 \). Wait, maybe the larger triangle's vertical side is \( y \), and horizontal side 22, and angle 44°, so \( \sin(44^\circ) = \frac{22}{z} \Rightarrow z = 22 / \sin(44^\circ) \approx 22 / 0.6947 \approx 31.7 \), no. Wait, the given \( z = 23.2 \). Let's check \( z = \sqrt{22^2 + y^2} \), if \( y = 7.3 \), then \( z = \sqrt{22^2 + 7.3^2} = \sqrt{484 + 53.29} = \sqrt{537.29} \approx 23.2 \). Ah! There we go. So the larger triangle has horizontal side 22, vertical side \( y \approx 7.3 \), right angle, so hypotenuse \( z = \sqrt{22^2 + y^2} \approx \sqrt{484 + 53.29} \approx \sqrt{537.29} \approx 23.2 \) (matches).

So summarizing:
- \( w \): from smaller triangle, \( \sin(44^\circ)=\frac{w}{12} \Rightarrow w = 12\sin(44^\circ) \approx 8.3 \).
- \( x \): from smaller triangle, \( \cos(44^\circ)=\frac{x}{12} \Rightarrow x = 12\cos(44^\circ) \approx 8.6 \).
- \( y \): from \( xy = 63 \Rightarrow y = 63 / x \approx 63 / 8.6 \approx 7.3 \).
- \( z \): from larger triangle (right triangle with legs 22 and \( y \)), \( z = \sqrt{22^2 + y^2} \approx \sqrt{484 + 53.29} \approx 23.2 \).

Answer:

\( w = 8.3 \), \( x = 8.6 \), \( y = 7.3 \), \( z = 23.2 \) (all rounded to nearest tenth, matching the given values after verification).